Arithmetic Structure in Dense Sets

这个项目主要集中在数论、组合学和遍历理论中的三个不同问题。这包括在加法组合学中关于算术级数的Szemerédi定理的推广（数字序列都是等距的，如4，6，8和10）的工作，非正式地说，任何足够大的整数集合都包含一个长的算术级数。在加法组合学中，确定“足够大”的大小是一个中心问题。研究人员将研究这个问题的版本，涉及比算术级数更复杂的模式，然后使用开发的结果和技术在遍历理论的相关问题上取得进展。研究者还将研究整数距离集的大小和结构，整数距离集是成对距离都是整数的点的集合。该奖项将支持本科生暑期在表示论和加法组合学方面的研究，并支持研究生的培训。更具体地说，研究人员将建立在她以前对缺乏不同次数多项式级数的整数子集和缺乏特定四点配置的有限域上的向量空间子集的定量界限的工作基础上，以解决更一般的多项式，多维，和多维多项式配置。多维多项式配置的不同程度的结果，然后将被用来取得进展的Furstenberg-Bergelson-Leibman猜想在遍历理论，其中涉及的逐点几乎处处收敛的某些非常规遍历平均。她还将调查的大小和结构的整数距离集，在欧几里德平面和更高的维度，通过编码它们作为子集的合理点对某些家庭的品种，然后研究这些品种。她将与她的本科生一起研究对称群特征表中的条目分布和高阶傅立叶分析中的一些算法问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

This project focuses primarily on three different problems in number theory, combinatorics, and ergodic theory. This includes work in additive combinatorics concerning generalizations of Szemerédi's theorem on arithmetic progressions (sequences of numbers that are all equally spaced, like 4, 6, 8, and 10), which, informally, says that any sufficiently large collection of whole numbers contains a long arithmetic progression. It is a central problem in additive combinatorics to determine how large "sufficiently large" is. The investigator will study versions of this question involving more complicated patterns than arithmetic progressions, and then use the results and techniques developed to make progress on a related problem in ergodic theory. The investigator will also study the size and structure of integer distance sets, which are sets of points whose pairwise distances are all whole numbers. This award will support undergraduate summer research on representation theory and additive combinatorics, and also support the training of graduate students.More specifically, the investigator will build on her previous work on quantitative bounds for subsets of the integers lacking polynomial progressions of distinct degrees and for subsets of vector spaces over finite fields lacking a certain four-point configuration to tackle more general polynomial, multidimensional, and multidimensional polynomial configurations. The results for multidimensional polynomial configurations of distinct degree will then be used to make progress on the Furstenberg--Bergelson--Leibman conjecture in ergodic theory, which concerns the pointwise almost everywhere convergence of certain nonconventional ergodic averages. She will also investigate the size and structure of integer distance sets, in both the Euclidean plane and in higher dimensions, by encoding them as subsets of rational points on certain families of varieties and then studying these varieties. With her undergraduate students, the investigator will study the distribution of entries in the character tables of symmetric groups and some algorithmic problems in higher-order Fourier analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.